Moving finite unit norm tight frames for $S^{n}$

Abstract

Frames for $\mathbb{R}^{n}$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word “frame” has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are $S^{1}$, $S^{3}$, and $S^{7}$. On the other hand, after combining the two separate meanings of the word “frame”, we show that the $n$-dimensional sphere, $S^{n}$, has a moving finite unit norm tight frame for its tangent bundle if and only if $n$ is odd. We give a procedure for creating vector fields on $S^{2n-1}$ for all $n\in\mathbb{N}$, and we characterize exactly when sets of such vector fields form a moving finite unit norm tight frame on $S^{2n-1}$. This gives as well a new method for constructing finite unit norm tight frames for Hilbert spaces.